<p>For studying the meromorphic degeneration of complex dynamics, the theory of hybrid spaces, introduced by Boucksom, Favre and Jonsson, is known to be a strong tool. In this paper, we apply this theory to the dynamics of Hénon maps. For a family of Hénon maps <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3726_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">$\{H_t\}_{t\in {\mathbb {D}}^}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>H</mi> <mi>t</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>t</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">D</mi> </mrow> <mo>∗</mo> </msup> </mrow> </msub> </math></EquationSource> </InlineEquation> that is parametrized by a unit punctured disk and meromorphically degenerates at the origin, we show that as <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3726_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$t\rightarrow 0,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <mn>0</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> the family of the invariant measures <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3726_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">$\{\mu _t\}_{t\in {\mathbb {D}}^}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>μ</mi> <mi>t</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>t</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">D</mi> </mrow> <mo>∗</mo> </msup> </mrow> </msub> </math></EquationSource> </InlineEquation> “weakly converges” to a measure on the Berkovich affine plane associated to the non-archimedean Hénon map determined by the family <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3726_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">$\{H_t\}_{t\in {\mathbb {D}}^*}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>H</mi> <mi>t</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>t</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">D</mi> </mrow> <mo>∗</mo> </msup> </mrow> </msub> </math></EquationSource> </InlineEquation>.</p>

Non-archimedean and hybrid dynamics of Hénon mappings

期刊论文
发表日期：

2025年5月3日

查看全文

Reimi Irokawa

摘要

For studying the meromorphic degeneration of complex dynamics, the theory of hybrid spaces, introduced by Boucksom, Favre and Jonsson, is known to be a strong tool. In this paper, we apply this theory to the dynamics of Hénon maps. For a family of Hénon maps $\{H_t\}_{t\in {\mathbb {D}}^*}$ ${H_{t}}_{t \in D^{*}}$ that is parametrized by a unit punctured disk and meromorphically degenerates at the origin, we show that as $t\rightarrow 0,$ $t \to 0,$ the family of the invariant measures $\{\mu _t\}_{t\in {\mathbb {D}}^*}$ ${μ_{t}}_{t \in D^{*}}$ “weakly converges” to a measure on the Berkovich affine plane associated to the non-archimedean Hénon map determined by the family $\{H_t\}_{t\in {\mathbb {D}}^*}$ ${H_{t}}_{t \in D^{*}}$ .